Classes d'homotopie de champs de vecteurs Morse-Smale sans singularit\'e sur les fibr\'es de Seifert

Abstract

In the first part of this paper, we consider smooth maps from a compact orientable 3-manifold without boundary to the 2-sphere. We give a geometric criterion to decide whether two given maps are homotopic, based on the sets of points where the maps are equal or antipodal. We extend this criterion to non-singular vector fields (and co-oriented plane fields) on 3-manifolds. In the second part, we make use of this criterion to study non-singular Morse-Smale vector fields on Seifert manifolds, giving a simple proof of a result of Yano.